Signed Real Hurwitz numbers

In this paper, we enumerate with signs the holomorphic maps between Real Riemann surfaces. We relate the signs and the numbers obtained to the Real Gromov-Witten theory of the target. For a finite group $G$ with a non-trivial morphism to $\{\pm 1\}$, we present a signed count of principal $G$-bundles yielding a Klein Topological Quantum Field Theory.